Non-Symmetric Jack Polynomials and Integral Kernels

Abstract

We investigate some properties of non-symmetric Jack, Hermite and Laguerre polynomials which occur as the polynomial part of the eigenfunctions for certain Calogero-Sutherland models with exchange terms. For the non-symmetric Jack polynomials, the constant term normalization ${\cal N}_\eta$ is evaluated using recurrence relations, and ${\cal N}_\eta$ is related to the norm for the non-symmetric analogue of the power-sum inner product. Our results for the non-symmetric Hermite and Laguerre polynomials allow the explicit determination of the integral kernels which occur in Dunkl's theory of integral transforms based on reflection groups of type $A$ and $B$, and enable many analogues of properties of the classical Fourier, Laplace and Hankel transforms to be derived. The kernels are given as generalized hypergeometric functions based on non-symmetric Jack polynomials. Central to our calculations is the construction of operators $\widehat{\Phi}$ and $\widehat{\Psi}$, which act as lowering-type operators for the non-symmetric Jack polynomials of argument $x$ and $x^2$ respectively, and are the counterpart to the raising-type operator $\Phi$ introduced recently by Knop and Sahi.